Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, use the formula for the sum of the first \(n\) terms of each geometric sequence, and then state the indicated sum. \(\sum_{n=1}^{9} 5 \cdot 2^{n-1}\)

Step-by-Step Solution

Answer

The sum of the first 9 terms is 2555.

Step 1: Identify the Geometric Sequence

The sequence can be identified as a geometric sequence with a common ratio and first term given. In this series, the first term is 5, and the common ratio is 2. This sequence comes from the general expression of a geometric sequence: \( a_n = ar^{n-1} \) where \( a = 5 \) and \( r = 2 \).

Step 2: Find the Formula for the Sum of a Geometric Sequence

The formula for the sum of the first \( n \) terms of a geometric sequence is: \[ S_n = a \frac{r^n - 1}{r - 1} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Step 3: Substitute Values into the Formula

Substitute \( a = 5 \), \( r = 2 \), and \( n = 9 \) into the formula: \[ S_9 = 5 \frac{2^9 - 1}{2 - 1} \].

Step 4: Calculate the Sum

First, calculate \( 2^9 \). \( 2^9 = 512 \). Substitute it back: \[ S_9 = 5 \frac{512 - 1}{1} = 5 \times 511 \].

Step 5: Finalize the Calculation

Multiply 5 by 511: \( 5 \times 511 = 2555 \). Thus, the sum of the first 9 terms of the sequence is 2555.

Key Concepts

Sum of Geometric Series

A geometric series is a sum of terms in a geometric sequence, where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio. When you want to find the total of the first few terms in a geometric sequence, you use the formula for the sum of a geometric series. This formula is useful because it allows you to calculate the sum without needing to add each term manually.
The sum of the first \( n \) terms of a geometric series is given by:
\[S_n = a \frac{r^n - 1}{r - 1}\]

• \( S_n \) is the sum of the series.
• \( a \) is the first term.
• \( r \) is the common ratio.
• \( n \) is the number of terms.

This formula is especially helpful when dealing with geometric sequences with a large number of terms, making calculations both quicker and error-free.

Common Ratio

The common ratio is a crucial element in understanding a geometric sequence. It represents the factor by which each term after the first is multiplied to produce the next term. Recognizing this pattern is key to identifying a sequence as geometric.
To find the common ratio \( r \), you divide any term in the sequence by the previous term. For example, in the sequence where the first few terms are 5, 10, 20,..., you calculate \( r \) as follows:
\[r = \frac{10}{5} = 2\]Knowing the common ratio allows one to predict future terms in the sequence and is vital for calculations involving sums of geometric series.

First Term

The first term, often denoted as \( a \), is the starting point of any geometric sequence. It is crucial for determining the entire sequence as every subsequent term is derived by multiplying the previous term by the common ratio.
In our exercise, the first term is 5. This means the sequence starts with 5, and each following term is a result of multiplying by the common ratio.
The first term not only helps set the basis for generating terms but also plays an important role in calculating the sum of the series. Without a known first term, it would be impossible to construct the full sequence or find its sum accurately.

Number of Terms

The number of terms \( n \) in a geometric sequence tells us how many terms are included in the geometric series that we are interested in or need to sum. Understanding \( n \) is important when calculating the sum using the sum formula.
In the given exercise, we see that \( n = 9 \), which means we are summing the first 9 terms of the sequence.
Knowing this number ensures that calculations are performed over the entire range of terms that you are concerned with, providing an accurate sum for the specified portion of the sequence.